Practice Questions

Q88.The number of matrices of order 3 × 3, whose entries are either 0 or 1 and the sum of all the entries is a prime number, is _______.

Q88.If the system of linear equations 2x −3y = γ + 5 αx + 5y = β + 1 , where α, β, γ ∈R has infinitely many solutions, then the value of |9α + 3β + 5γ| is equal to

Q88.Let M and N be the number of points on the curve y5 −9xy + 2x = 0 , where the tangents to the curve are parallel to x-axis and y-axis, respectively. Then the value of M + N equals _______.

Q88.Let S be the region bounded by the curves y = x3 and y2 = x. The curve y = 2|x| divides S into two regions of areas R1 and R2 . If max|R1, R2| = R2 , then R1R2 is equal to ______

Q88.For real numbers a, b(a > b > 0), let x2 y2 = 30π Area {(x, y) : x2 + y2 ≤a2 and a2 + b2 ≥1} and x2 y2 = 18π Area {(x, y) : x2 + y2 ≥b2 and a2 + b2 ≤1} Then the value of (a −b)2 is equal to _____.

Q88.Let the tangents at the points P and Q on the ellipse x2 S is 2 + 4 = 1 meet at the point R(√2, 2√2 −2). If the focus of the ellipse on its negative major axis, then SP 2 + SQ2 is equal to π dx is equal to

Q88.Let S = {(−10 ab ); 100 elements in n=1Tn∩ is _____.

Q88.If the tangent to the curve 𝑦= 𝑥3 - 𝑥2 + 𝑥 at the point 𝑎, 𝑏 is also tangent to the curve 𝑦= 5𝑥2 + 2x - 25 at the point 2, - 1, then 2𝑎+ 9𝑏 is equal to ______. 2 2 2 2 2

Q89.Let l be a line which is normal to the curve y = 2x2 + x + 2 at a point P on the curve. If the point Q(6, 4) lies on the line l and O is origin, then the area of the triangle OPQ is equal to _____. → →

Q89.The area (in sq. units) of the region enclosed between the parabola y2 = 2x and the line x + y = 4 is ______.

Q89.Let d be the distance between the foot of perpendiculars of the points P(1, 2 −1) and Q(2, −1, 3) on the plane −x + y + z = 1 . Then d2 is equal to ______. JEE Main 2022 (29 Jun Shift 1) JEE Main Previous Year Paper = 4 be a plane. Let P2 be another plane which passes through the points

Q89.Let a line having direction ratios 1, −4, 2 intersect the lines x−73 = y−1−1 = z+21 and x2 = y−73 = 1z at the points A and B. Then (AB)2 is equal to + + = + + +

Q89.If 𝛼√2 + 𝛽√3, where 𝛼, 𝛽 are integers, then 𝛼+ 𝛽 is equal to ∫0 √1 + 𝑥2 + √1 + 𝑥23𝑑𝑥= 56 43 111

Q89.Let 𝜃 be the angle between the vectors →𝑎 and →𝑏, where →𝑎= 4, →𝑏= 3 and 𝜃∈𝜋 𝜋 Then 4, 3. 2 2 →𝑎- →𝑏× →𝑎+ →𝑏 + 4→𝑎· →𝑏 is equal to ______

Q89.The integral 24 is equal to ______. π ∫ 0 (2+x2)√4+x4

Q89.Let the solution curve y = y(x) of the differential equation (4 + x2)dy −2x(x2 + 3y + 4)dx = 0 pass through the origin. Then y(2) is equal to _____.

Q89.If 𝑙𝑖𝑚 lim ( 𝑛𝑘+ 1 ) + ( 𝑛𝑘+ 2 ) + … + 𝑛𝑘+ 𝑛= 33 . 1 1 · 1𝑘+ 2𝑘+ 3𝑘+ … + 𝑛𝑘, then the 𝑛𝑘+ 𝑛𝑘+ 𝑛→∞ 𝑛→∞ integral value of 𝑘 is equal to _____ . 𝑥- 2 𝑦- 1 𝑧 𝑥- 3 𝑦- 5 𝑧- 1

Q89.Let →𝑎 and →𝑏 be two vectors such that →𝑎+ →𝑏 = →𝑎 + 2 →𝑏 , →𝑎· →𝑏= 3 and →𝑎× →𝑏 = 75. Then →𝑎 is equal to ______.

Q89.Let an = ∫n−1(1 + x2 + x23 + … + xn−1n )dx for every {n ∈N : an ∈(2, 30)} is _________. , y(1) = 1. If for some

Q89.Let A1 = {(x, y) : |x| ≤y2, |x| + 2y ≤8} and A2 = {(x, y) : |x| + |y| ≤k}. If 27 (Area A1 ) = 5 (Area A2 ), then k is equal to

Q89.Let a curve y = y(x) pass through the point (3, 3) and the area of the region under this curve, above the x-axis y 3 and between the abscissae 3 and x(> 3) be ( x ) . If this curve also passes through the point (α, 6√10) in the first quadrant, then α is equal to _______. y+2

Q89.Let y = y(x), x > 1 , be the solution of the differential equation (x −1) dxdy + 2xy = x−11 , with y(2) = 1+e42e4 . If y(3) = eα+1βeα . then the value of α + β is equal to ______. → → , then the value of is b 3(→c.→a)

Q89.The value of the integral ∫ 0 2 60 sin(6x)sin x JEE Main 2022 (28 Jul Shift 2) JEE Main Previous Year Paper

Q90.A bag contains 4 white and 6 black balls. Three balls are drawn at random from the bag. Let X be the number of white balls, among the drawn balls. If σ2 is the variance of X , then 100σ2 is equal to JEE Main 2022 (28 Jul Shift 2) JEE Main Previous Year Paper

Q90.Let the image of the point P(1, 2, 3) in the line L : x−63 = y−12 = z−23 be Q. let R(α, β, γ) be a point that divides internally the line segment PQ in the ratio 1 : 3 . Then the value of 22(α + β + γ) is equal to JEE Main 2022 (28 Jun Shift 2) JEE Main Previous Year Paper